This page will show the issues and challenges we have identified for learners at various stages through the sub strand of Number and Place Value. These issues and challenges will be linked to information, solutions, explanations and examples that we have identified through course literature and our personal experience with learners.
The Concept of Zero
The concept of zero causes difficulties for children, not only in reading and writing larger numbers but because the notion of 'writing something for nothing' itself seems odd. Zero can be viewed as nothing rather than a number representing none of something. This is further reinforced when writing numbers in words, 123 for example is written as one hundred and twenty-three, each number has a word to represent it, this is not true for numbers such as 103, this number is written as one hundred and three, the zero has no mention in the words and so is ignored by students.
Materials don't always assist because when there are no tens and hundreds there is nothing to see with materials either. Care must be taken in sequencing the development of the number names and the number symbols to allow understanding of the lack of particular units and the use of zero to show there is none of that place (Booker et al, 2004, pg 79, 80, 93).
It has been said the the concept of zero can be attributed to the Hindus. They used the zero in the way it is used today. The symbol of 0 comes from the Hindus also. Arabs spread the Hindu decimal zero and its new mathematics to Europe in the Middle Ages (Interesting Maths History, n.d.).
Activity: Students sit in a circle, teacher directs the discussion and questioning. Ask a student to jump 5 (five) times, have the other students count as the student jumps. Ask another child to jump 0 (zero) times. When the student doesn't jump ask "Why didnt you do anything?" Ask the rest of the class if they agree. Discuss with the class why this student did not jump. Ensure that zero is viewed as a integer also (Montessori Primary Guide).
Find some places where people such as your family use numbers. Look for numbers where zero is used but has no meaning. One example of this is the odometer on a car where zero appears in front of the total kilometres travelled (Sullivan and Lilburn, 2006, pg 35).
The use of a place value chart can also be used to correct students misunderstandings about the value zero. A blank chart is given to students along with concrete materials to represent numbers. Ask students to represent 13 (thirteen) on one chart and then 103 (one hundred and three) on another chart with the concrete materials and cards with the digits (deliberately leaving out the digit zero). Move the digits off the place value chart and ask students to look at the numbers and say what they see. Students should say that they see a 1 and a 3 or 13 for both numbers. Ask students what they notice about the two sets of numbers and if they match the representations on the place value chart. Students should identify a difference between the numbers and their representations on the place value chart. Next ask the students to move the numbers back to the place value chart asking them to name the value of the digits in relation to their place value. For 13 the three represents 3 ones and the 1 represents 1 ten so these digits are placed on the place value chart. For 103 the three represents 3 ones and the 1 represents 1 hundred, these digits are placed on the place value chart, here the teacher introduces the zero digit card and explicitly teaches that where there is no value in a place the digit zero must be written as it represents a value. The use of unbundeling hundreds and rebundleing hundres so they are able to see that the 0 is represented.
The understanding of single digit numbers
Difficulties with one-digit numbers occur when children have not linked number names and symbols to the objects they represent. This shows up in their counting and writing of the digits (Booker et al, 2004, pg 93). Students need a sense of number before they are introduced to the symbols of numbers. Number sense is not a body of knowledge but a way of thinking. (Siemens et al, 2011, pg 187).
Students may struggle with the digits 7, 2, 3 and 5 and these digits may often reverse and written backwards. Children may also experience confusion with the digits 6 and 9 or 2 and 5.
Activities: Involving the students in making sets of number cards can be a way of allowing the students to see and practice writing the numbers in numerical form as well as seeing the concrete and written number in letters. These are then displayed in the classroom.
Other activities for display on the walls are tracing games and rhymes. 1. Number one starts at the top. Straight down we go until we stop. 2. Around, down, across we go. Number 2 is made just so. 3. Around, around like a little bee. Sniffing the flowers in number three. 4. Straight down to near the line. Turn rignt and cross so. Four is fine. 5. Number 5's head is first, his body's fat. Don't forget on top-his hat. 6. Straight little back and big round tummy. Number six looks oh so funny. 7. Left to right and down we zoom. Seven is standing on the room. 8. Make an 'S' for number 8 and do not stop till you reach to top. 9. Along and around, up and down. Number nine is a real clown. (Siemens et al, 2011, p. 283).
Place value
Difficulties with an algorithm usually have their origins with appropriate renaming particular misconceptions with zero, or confusion with the recording of their working because of an inability to use place value. (Booker et al, 2004, pg 210).
Place Value: Adds all the digits without regard to place value 25 +32 12 When numbers are used from a problem, places may not be aligned 382 +46 842 Activity The use of concrete materials to demonstrate and teach place value will help student's understanding. What numbers can you make using 6, 5 and 8? Do children record their answers systematically and know when they have recorded all possibilities? Do children record their answers systematically and know when they have recorded all possibilities? Do any children include single digits, e.g 5, as one of their answers (Sullivan and Lilburn, 2006, pg 32).
Understanding of teen numbers
While teen numbers are written and represented with materials in exactly the same way as the numbers 20-99, pronunciation of the teen number words and the order of the symbols representing them has hindered children to say and write them incorrectly, for example writing 81 for eighteen, or confusing eighteen with 80. If the pattern of the two digit number had been followed, a number such as 19 would be said 'one-t-nine' rather than the back to front nineteen. Also the numbers eleven and twelve give no indication of their tens and ones structure (Booker et al, 2004, pg 93).
The development of teen numbers is best left until last when the focus can be on the irregular language that is the source of the difficulty (Booker et al, 2004, pg 93).
Activity Rhyme:A 1 and a 1 make 11 funA 1 and a 2 tell twelve what to doA 1 and a 3 send 13 up a treeA 1 and a 4 make 14 shut the doorA 1 and a 5 keep 15 aliveA 1 and a 6 make 16 pick up sticksA 1 and a 7 send 17 to heavenA 1 and an 8 make 18 greatA 1 and a 9 make 19 shineA 2 and a 0 make 20 the hero Action suggestions: A 1 (hold up one pointer) and a 1 (hold up the other pointer) make 11 fun (do a dance) A 1 (put up one pointer) and a 2 (hold up 2 fingers on the other hand) tell twelve what to do (I put one hand by my elbow on the other arm like I'm shaking a bossy pointer finger at someone and use a bossy tone.) A one (one pointer) and a three (three fingers on the other hand) send 13 up a tree (I bring the one and the 3 hand together and push them skywards.) A one (hold up 1 pointer) and a 4 (hold up the other 4 fingers) make 14 shut the door. (I bring the two hands together like a door shutting) A one (hold up one pointer) and a 5 (hold up the other 5) keep 15 alive (I pat my 5 hand on my chest like a heart beating.) A one (one finger and then you need to get creative I put the one away and hold up 6) and a 6 make 16 pick up sticks (We reach out and pretend we are grabbing sticks) A one and a 7 send 17 to heaven (we just point skyward) A one and an 8 make 18 great (is done like a cheerleader arms) A one (they put one spread hand by their face thumb touching their cheek) and a 9 the other hand on the other side) make 19 shine (because I always tell them I love to see their shining faces and that they are my shining stars. One kid said this must be your favorite number :-) and now it is :-) A 2 (make a muscle with one arm) and a zero (make the other arm into a muscle) make 20 the hero (we use a hero strong voice and pull our muscles up and down) http://www.mrsjonesroom.com/songs/teenchant.html This activity would be for children in Foundation to Year 2
The value of decimals
Difficulties with decimals usually occur because place value is not used in naming the numbers, and they are spelled out as separate digit, for example 0.25 may be said 'zero point two five' rather than read 'twenty five hundredths'. There may be confusion about the new decimal places and the earlier whole number places. (Booker et al, 2004, pg 151). Misconceptions can also arise from the length of the decimal, for example 5.03 might be thought of as being larger than 5.3 simply because there are more numbers in the string, (Siemon, 2011). One way of counteracting this is to encourage students to consider the amount of decimal places by the type of mathematical inquiry it is. Do not stick to the rounding to two decimal places as is stated in most school texts. (Siemon, 2011).
Activity Tenths mats can be used to teach the beginnings of decimal recording. Students can also use straws and number lines to portray a concrete version of decimals. (Siemon, 2011, p.425) This is an example of a tenths mat, students have visual representations of 1/10th as a fraction and a decimal. this is a coloured version 4/10ths or 0.4 of the mat coloured in.
Use of repeated addition instead of multiplication
Often children endeavour simply to memorise all of the answers for multiplication facts, and then struggle to search through these numbers to find the appropriate result. As a consequence, children ask questions such as 'does 6 multiplied by 9 equal 56 or 54?' when they confuse the answers for 7 eights (56) and 6 nines (54) which are almost the same. Some strategies children use are counting on, using their fingers, and sometimes dots on a page or tapping. Other children may use doubling combined with addition or subtraction to adjust their result but lose track of what they are trying to do. If there is confusion with addition answers, this usually means that the multiplication concept is not secure and will need to be built up to show what multiplication is and provide a basis for building multiplication facts appropriately.
A child may guess or count on his or her fingers to obtain answers rather than use more appropriate thinking. For example, a fact such as '6 eights' may be answered '8, 16, 24, 32, 40, 48', using fingers to keep track of the number of times 8 is added. Not only is this inefficient, it shows that the multiplication concept is focused on the very early repeated addition notion. (Booker et al, 2004, pg 274)
This becomes an issue as students do not see multiplication as a more efficient way of solving repeated addition problems. This usually develops when students are not given an opportunity to represent their thinking in different ways.
Activity Think boards are useful tools to encourage and enable students to solve problems using different strategies. A think board is an A4 piece of paper divided into four sections with the main concept to be explored written in the middle of the page. The four sections can be labelled with anything that suits the activity. Suggestions for this issue would be ‘Concrete Materials’, ‘Number Sentences’, ‘Visual’ and ‘Story Problems’. Students work through the four sections to represent the number or concept in the middle. Alternatively the middle box can be left empty and the learning manager can fill in one of the four sections and the students fill in the rest. For example the teacher might write a story problem that says there were four plates with five cakes on each plate; students must then fill in the other three sections. This allows students to represent and see the concept in different ways. In the number sentence section students may write 5+5+5+5 or 4 plates of 5 or 4 groups of 5 and then 4 x 5.
Understanding of index notation
When the same factor occurs several times a more concise notation using notation using exponents is used to record the product; for example, 3X3X3X3 is recorded 34. However, because many children confuse the use of exponents with multiplication, their introduction should not occur until large numbers are familiar, multiplication itself is very secure and children see the need and power of this new way of recording numbers. The first step to build up this new notation is to develop the idea of exponent as a shortened means of recording the result of multiplying the same factor several times (Booker et al, 2004, pg 129).
The correct way to see 4² is 4x4 = 16 In order to correct this challenge learners need to use hands on materials. Use a set of small cubes two different colours to represent 4x4 for learners. This makes a square for learners to easily see, so they will understand that 42 is also called 4 squared.
This is the same for 43 This is often seen as 4 x 3 = 12 this is incorrect. The correct way to see this 43 is 4x4x4=64 Once again this can be explained and understood using hands on materials. Use a set of cubes in four different colours to represent 4x4x4=64 for learners. This makes a cube for learners to easily see, so they will understand that 43 is also call 4 cubed.
Estimation
According to Siemen et al, (2011, p 541), estimation is the process of arriving at an inexact result on the basis of general considerations of the numbers and operations involved as opposed to a result based on a precise mathematical procedure. In order to estimate students must have a well-developed sense of number, knowledge of number facts and relationships. Students also require a sound knowledge of operations. The misconception around estimation is that estimation is simply guessing an answer to a problem. Estimation is actually a complex skill. In order to estimate a reasonable answer students require information about the problem. The learning manager must provide students with many opportunities to export estimation and ways to arrive at a reasonable estimation for problems. (pg 388, 389) Some strategies to be taught include multiplication, division and fractioning. Activites: One activity to use in the classroom could be rounding problems. These are real life situations where the student is given some information and some is unknown allowing the student space for guess work. An example might be; Mrs Patel is driving to a town 143 km away. The speed limit varies between 50 and 60 km/hr. There are some road works along the way. How much time do you think she should give herself to make the journey? Give reasons for your answers and show any calculations. (Australian curriculum mathematics resource book, 2012) Problems that may arise from this type of problem is the language might be difficult for student to understand. Some students may need extra support with reading and picking out the relevant information, (Siemen, 2011 p.563).
This page will show the issues and challenges we have identified for learners at various stages through the sub strand of Number and Place Value. These issues and challenges will be linked to information, solutions, explanations and examples that we have identified through course literature and our personal experience with learners.
The Concept of Zero
The concept of zero causes difficulties for children, not only in reading and writing larger numbers but because the notion of 'writing something for nothing' itself seems odd. Zero can be viewed as nothing rather than a number representing none of something.This is further reinforced when writing numbers in words, 123 for example is written as one hundred and twenty-three, each number has a word to represent it, this is not true for numbers such as 103, this number is written as one hundred and three, the zero has no mention in the words and so is ignored by students.
Materials don't always assist because when there are no tens and hundreds there is nothing to see with materials either. Care must be taken in sequencing the development of the number names and the number symbols to allow understanding of the lack of particular units and the use of zero to show there is none of that place (Booker et al, 2004, pg 79, 80, 93).
It has been said the the concept of zero can be attributed to the Hindus. They used the zero in the way it is used today. The symbol of 0 comes from the Hindus also. Arabs spread the Hindu decimal zero and its new mathematics to Europe in the Middle Ages (Interesting Maths History, n.d.).
Activity:
Students sit in a circle, teacher directs the discussion and questioning.
Ask a student to jump 5 (five) times, have the other students count as the student jumps. Ask another child to jump 0 (zero) times. When the student doesn't jump ask "Why didnt you do anything?" Ask the rest of the class if they agree. Discuss with the class why this student did not jump. Ensure that zero is viewed as a integer also (Montessori Primary Guide).
Find some places where people such as your family use numbers. Look for numbers where zero is used but has no meaning. One example of this is the odometer on a car where zero appears in front of the total kilometres travelled (Sullivan and Lilburn, 2006, pg 35).
The use of a place value chart can also be used to correct students misunderstandings about the value zero. A blank chart is given to students along with concrete materials to represent numbers. Ask students to represent 13 (thirteen) on one chart and then 103 (one hundred and three) on another chart with the concrete materials and cards with the digits (deliberately leaving out the digit zero). Move the digits off the place value chart and ask students to look at the numbers and say what they see. Students should say that they see a 1 and a 3 or 13 for both numbers. Ask students what they notice about the two sets of numbers and if they match the representations on the place value chart. Students should identify a difference between the numbers and their representations on the place value chart. Next ask the students to move the numbers back to the place value chart asking them to name the value of the digits in relation to their place value. For 13 the three represents 3 ones and the 1 represents 1 ten so these digits are placed on the place value chart. For 103 the three represents 3 ones and the 1 represents 1 hundred, these digits are placed on the place value chart, here the teacher introduces the zero digit card and explicitly teaches that where there is no value in a place the digit zero must be written as it represents a value.
The use of unbundeling hundreds and rebundleing hundres so they are able to see that the 0 is represented.
The understanding of single digit numbers
Difficulties with one-digit numbers occur when children have not linked number names and symbols to the objects they represent. This shows up in their counting and writing of the digits (Booker et al, 2004, pg 93). Students need a sense of number before they are introduced to the symbols of numbers. Number sense is not a body of knowledge but a way of thinking. (Siemens et al, 2011, pg 187).Students may struggle with the digits 7, 2, 3 and 5 and these digits may often reverse and written backwards. Children may also experience confusion with the digits 6 and 9 or 2 and 5.
Activities:
Involving the students in making sets of number cards can be a way of allowing the students to see and practice writing the numbers in numerical form as well as seeing the concrete and written number in letters. These are then displayed in the classroom.
Other activities for display on the walls are tracing games and rhymes.1. Number one starts at the top. Straight down we go until we stop.
2. Around, down, across we go. Number 2 is made just so.
3. Around, around like a little bee. Sniffing the flowers in number three.
4. Straight down to near the line. Turn rignt and cross so. Four is fine.
5. Number 5's head is first, his body's fat. Don't forget on top-his hat.
6. Straight little back and big round tummy. Number six looks oh so funny.
7. Left to right and down we zoom. Seven is standing on the room.
8. Make an 'S' for number 8 and do not stop till you reach to top.
9. Along and around, up and down. Number nine is a real clown.
(Siemens et al, 2011, p. 283).
Place value
Difficulties with an algorithm usually have their origins with appropriate renaming particular misconceptions with zero, or confusion with the recording of their working because of an inability to use place value. (Booker et al, 2004, pg 210).Place Value: Adds all the digits without regard to place value 25
+32
12
When numbers are used from a problem, places may not be aligned 382
+46
842
Activity
The use of concrete materials to demonstrate and teach place value will help student's understanding.
What numbers can you make using 6, 5 and 8? Do children record their answers systematically and know when they have recorded all possibilities? Do children record their answers systematically and know when they have recorded all possibilities? Do any children include single digits, e.g 5, as one of their answers (Sullivan and Lilburn, 2006, pg 32).
Understanding of teen numbers
While teen numbers are written and represented with materials in exactly the same way as the numbers 20-99, pronunciation of the teen number words and the order of the symbols representing them has hindered children to say and write them incorrectly, for example writing 81 for eighteen, or confusing eighteen with 80. If the pattern of the two digit number had been followed, a number such as 19 would be said 'one-t-nine' rather than the back to front nineteen. Also the numbers eleven and twelve give no indication of their tens and ones structure (Booker et al, 2004, pg 93).The development of teen numbers is best left until last when the focus can be on the irregular language that is the source of the difficulty (Booker et al, 2004, pg 93).
Activity
Rhyme:A 1 and a 1 make 11 funA 1 and a 2 tell twelve what to doA 1 and a 3 send 13 up a treeA 1 and a 4 make 14 shut the doorA 1 and a 5 keep 15 aliveA 1 and a 6 make 16 pick up sticksA 1 and a 7 send 17 to heavenA 1 and an 8 make 18 greatA 1 and a 9 make 19 shineA 2 and a 0 make 20 the hero
Action suggestions:
A 1 (hold up one pointer) and a 1 (hold up the other pointer) make 11 fun (do a dance)
A 1 (put up one pointer) and a 2 (hold up 2 fingers on the other hand) tell twelve what to do (I put one hand by my elbow on the other arm like I'm shaking a bossy pointer finger at someone and use a bossy tone.)
A one (one pointer) and a three (three fingers on the other hand) send 13 up a tree (I bring the one and the 3 hand together and push them skywards.)
A one (hold up 1 pointer) and a 4 (hold up the other 4 fingers) make 14 shut the door. (I bring the two hands together like a door shutting)
A one (hold up one pointer) and a 5 (hold up the other 5) keep 15 alive (I pat my 5 hand on my chest like a heart beating.)
A one (one finger and then you need to get creative I put the one away and hold up 6) and a 6 make 16 pick up sticks (We reach out and pretend we are grabbing sticks)
A one and a 7 send 17 to heaven (we just point skyward)
A one and an 8 make 18 great (is done like a cheerleader arms)
A one (they put one spread hand by their face thumb touching their cheek) and a 9 the other hand on the other side) make 19 shine (because I always tell them I love to see their shining faces and that they are my shining stars. One kid said this must be your favorite number :-) and now it is :-)
A 2 (make a muscle with one arm) and a zero (make the other arm into a muscle) make 20 the hero (we use a hero strong voice and pull our muscles up and down)
http://www.mrsjonesroom.com/songs/teenchant.html
This activity would be for children in Foundation to Year 2
The value of decimals
Difficulties with decimals usually occur because place value is not used in naming the numbers, and they are spelled out as separate digit, for example 0.25 may be said 'zero point two five' rather than read 'twenty five hundredths'. There may be confusion about the new decimal places and the earlier whole number places. (Booker et al, 2004, pg 151).Misconceptions can also arise from the length of the decimal, for example 5.03 might be thought of as being larger than 5.3 simply because there are more numbers in the string, (Siemon, 2011).
One way of counteracting this is to encourage students to consider the amount of decimal places by the type of mathematical inquiry it is. Do not stick to the rounding to two decimal places as is stated in most school texts. (Siemon, 2011).
Activity
Tenths mats can be used to teach the beginnings of decimal recording. Students can also use straws and number lines to portray a concrete version of decimals. (Siemon, 2011, p.425)
Use of repeated addition instead of multiplication
Often children endeavour simply to memorise all of the answers for multiplication facts, and then struggle to search through these numbers to find the appropriate result. As a consequence, children ask questions such as 'does 6 multiplied by 9 equal 56 or 54?' when they confuse the answers for 7 eights (56) and 6 nines (54) which are almost the same. Some strategies children use are counting on, using their fingers, and sometimes dots on a page or tapping. Other children may use doubling combined with addition or subtraction to adjust their result but lose track of what they are trying to do. If there is confusion with addition answers, this usually means that the multiplication concept is not secure and will need to be built up to show what multiplication is and provide a basis for building multiplication facts appropriately.A child may guess or count on his or her fingers to obtain answers rather than use more appropriate thinking. For example, a fact such as '6 eights' may be answered '8, 16, 24, 32, 40, 48', using fingers to keep track of the number of times 8 is added. Not only is this inefficient, it shows that the multiplication concept is focused on the very early repeated addition notion.
(Booker et al, 2004, pg 274)
This becomes an issue as students do not see multiplication as a more efficient way of solving repeated addition problems. This usually develops when students are not given an opportunity to represent their thinking in different ways.
Activity
Think boards are useful tools to encourage and enable students to solve problems using different strategies. A think board is an A4 piece of paper divided into four sections with the main concept to be explored written in the middle of the page. The four sections can be labelled with anything that suits the activity. Suggestions for this issue would be ‘Concrete Materials’, ‘Number Sentences’, ‘Visual’ and ‘Story Problems’. Students work through the four sections to represent the number or concept in the middle. Alternatively the middle box can be left empty and the learning manager can fill in one of the four sections and the students fill in the rest. For example the teacher might write a story problem that says there were four plates with five cakes on each plate; students must then fill in the other three sections. This allows students to represent and see the concept in different ways. In the number sentence section students may write 5+5+5+5 or 4 plates of 5 or 4 groups of 5 and then 4 x 5.
Understanding of index notation
When the same factor occurs several times a more concise notation using notation using exponents is used to record the product; for example, 3X3X3X3 is recorded 34. However, because many children confuse the use of exponents with multiplication, their introduction should not occur until large numbers are familiar, multiplication itself is very secure and children see the need and power of this new way of recording numbers. The first step to build up this new notation is to develop the idea of exponent as a shortened means of recording the result of multiplying the same factor several times (Booker et al, 2004, pg 129).The correct way to see 4² is 4x4 = 16
In order to correct this challenge learners need to use hands on materials.
Use a set of small cubes two different colours to represent 4x4 for learners. This makes a square for learners to easily see, so they will understand that 42 is also called 4 squared.
This is the same for 43
This is often seen as 4 x 3 = 12 this is incorrect.
The correct way to see this 43 is 4x4x4=64
Once again this can be explained and understood using hands on materials.
Use a set of cubes in four different colours to represent 4x4x4=64 for learners. This makes a cube for learners to easily see, so they will understand that 43 is also call 4 cubed.
Estimation
According to Siemen et al, (2011, p 541), estimation is the process of arriving at an inexact result on the basis of general considerations of the numbers and operations involved as opposed to a result based on a precise mathematical procedure.In order to estimate students must have a well-developed sense of number, knowledge of number facts and relationships. Students also require a sound knowledge of operations. The misconception around estimation is that estimation is simply guessing an answer to a problem. Estimation is actually a complex skill. In order to estimate a reasonable answer students require information about the problem. The learning manager must provide students with many opportunities to export estimation and ways to arrive at a reasonable estimation for problems. (pg 388, 389)
Some strategies to be taught include multiplication, division and fractioning.
Activites:
One activity to use in the classroom could be rounding problems. These are real life situations where the student is given some information and some is unknown allowing the student space for guess work.
An example might be;
Mrs Patel is driving to a town 143 km away. The speed limit varies between 50 and 60 km/hr. There are some road works along the way.
How much time do you think she should give herself to make the journey? Give reasons for your answers and show any calculations.
(Australian curriculum mathematics resource book, 2012)
Problems that may arise from this type of problem is the language might be difficult for student to understand. Some students may need extra support with reading and picking out the relevant information, (Siemen, 2011 p.563).